mmtheorems63 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 6201-6300 - Page 63 of 175

Type	Label	Description
Statement
Theorem	nqex 6201	The class of positive fractions exists.
|- Q. e. _V
Theorem	0npq 6202	The empty set is not a positive fraction.
|- -. (/) e. Q.
Theorem	ltrelpq 6203	Positive fraction 'less than' is a relation on positive fractions.
|- <Q C_ (Q. X. Q.)
Theorem	addcmpblnq 6204	Lemma showing compatibility of addition.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- D e. _V   &   |- F e. _V   &   |- G e. _V   &   |- R e. _V   &   |- S e. _V   =>   |- ((((A e. N. /\ B e. N.) /\ (C e. N. /\ D e. N.)) /\ ((F e. N. /\ G e. N.) /\ (R e. N. /\ S e. N.))) -> (((A .N D) = (B .N C) /\ (F .N S) = (G .N R)) -> <.((A .N G) +N (B .N F)), (B .N G)>. ~Q <.((C .N S) +N (D .N R)), (D .N S)>.))
Theorem	mulcmpblnq 6205	Lemma showing compatibility of multiplication.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- D e. _V   &   |- F e. _V   &   |- G e. _V   &   |- R e. _V   &   |- S e. _V   =>   |- ((((A e. N. /\ B e. N.) /\ (C e. N. /\ D e. N.)) /\ ((F e. N. /\ G e. N.) /\ (R e. N. /\ S e. N.))) -> (((A .N D) = (B .N C) /\ (F .N S) = (G .N R)) -> <.(A .N F), (B .N G)>. ~Q <.(C .N R), (D .N S)>.))
Theorem	addpipq 6206	Addition of positive fractions in terms of positive integers.
|- (((A e. N. /\ B e. N.) /\ (C e. N. /\ D e. N.)) -> ([<.A, B>.] ~Q +Q [<.C, D>.] ~Q ) = [<.((A .N D) +N (B .N C)), (B .N D)>.] ~Q )
Theorem	mulpipq 6207	Multiplication of positive fractions in terms of positive integers.
|- (((A e. N. /\ B e. N.) /\ (C e. N. /\ D e. N.)) -> ([<.A, B>.] ~Q .Q [<.C, D>.] ~Q ) = [<.(A .N C), (B .N D)>.] ~Q )
Theorem	ordpipq 6208	Ordering of positive fractions in terms of positive integers.
|- A e. _V   &   |- B e. _V   &   |- C e. _V   &   |- D e. _V   =>   |- ([<.A, B>.] ~Q <Q [<.C, D>.] ~Q <-> (A .N D) <N (B .N C))
Theorem	1q 6209	The positive fraction 'one'.
|- 1Q e. Q.
Theorem	addclpq 6210	Closure of addition on positive fractions.
|- ((A e. Q. /\ B e. Q.) -> (A +Q B) e. Q.)
Theorem	dmaddpq 6211	Domain of addition on positive fractions.
|- dom +Q = (Q. X. Q.)
Theorem	mulclpq 6212	Closure of multiplication on positive fractions.
|- ((A e. Q. /\ B e. Q.) -> (A .Q B) e. Q.)
Theorem	dmmulpq 6213	Domain of multiplication on positive fractions.
|- dom .Q = (Q. X. Q.)
Theorem	addcompq 6214	Addition of positive fractions is commutative.
|- A e. _V   &   |- B e. _V   =>   |- (A +Q B) = (B +Q A)
Theorem	addasspq 6215	Addition of positive fractions is associative.
|- B e. _V   &   |- C e. _V   =>   |- ((A +Q B) +Q C) = (A +Q (B +Q C))
Theorem	mulcompq 6216	Multiplication of positive fractions is commutative.
|- A e. _V   &   |- B e. _V   =>   |- (A .Q B) = (B .Q A)
Theorem	mulasspq 6217	Multiplication of positive fractions is associative.
|- B e. _V   &   |- C e. _V   =>   |- ((A .Q B) .Q C) = (A .Q (B .Q C))
Theorem	distrpqlem 6218	Lemma for distributive law: cancellation of common factor.
Theorem	distrpq 6219	Multiplication of positive fractions is distributive.
|- B e. _V   &   |- C e. _V   =>   |- (A .Q (B +Q C)) = ((A .Q B) +Q (A .Q C))
Theorem	1qec 6220	The equivalence class of ratio 1.
|- A e. _V   =>   |- (A e. N. -> 1Q = [<.A, A>.] ~Q )
Theorem	mulidpq 6221	Multiplication identity element for positive fractions.
|- (A e. Q. -> (A .Q 1Q) = A)
Theorem	recmulpq 6222	Relationship between reciprocal and multiplication on positive fractions.
|- B e. _V   =>   |- (A e. Q. -> ((*Q` A) = B <-> (A .Q B) = 1Q))
Theorem	recidpq 6223	A positive fraction times its reciprocal is 1.
|- (A e. Q. -> (A .Q (*Q` A)) = 1Q)
Theorem	recclpq 6224	Closure law for positive fraction reciprocal.
|- (A e. Q. -> (*Q` A) e. Q.)
Theorem	recrecpq 6225	Reciprocal of reciprocal of positive fraction.
|- A e. _V   =>   |- (A e. Q. -> (*Q` (*Q` A)) = A)
Theorem	dmrecpq 6226	Domain of reciprocal on positive fractions.
|- dom *Q = Q.
Theorem	ltsopq 6227	'Less than' is a strict ordering on positive fractions.
|- <Q Or Q.
Theorem	ltapq 6228	Ordering property of addition for positive fractions. Proposition 9-2.6(ii) of [Gleason] p. 120.
|- A e. _V   &   |- B e. _V   =>   |- (C e. Q. -> (A <Q B <-> (C +Q A) <Q (C +Q B)))
Theorem	ltmpq 6229	Ordering property of multiplication for positive fractions. Proposition 9-2.6(iii) of [Gleason] p. 120.
|- A e. _V   &   |- B e. _V   =>   |- (C e. Q. -> (A <Q B <-> (C .Q A) <Q (C .Q B)))
Theorem	1lt2pq 6230	One is less than two (one plus one).
|- 1Q <Q (1Q +Q 1Q)
Theorem	ltaddpq 6231	The sum of two fractions is greater than one of them.
|- A e. _V   &   |- B e. _V   =>   |- ((A e. Q. /\ B e. Q.) -> A <Q (A +Q B))
Theorem	ltexpq 6232	Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119.
|- A e. _V   =>   |- ((A e. Q. /\ B e. Q.) -> (A <Q B <-> E.x(A +Q x) = B))
Theorem	ltexpq2 6233	Ordering on positive fractions in terms of existence of sum. Definition in Proposition 9-2.6 of [Gleason] p. 119.
|- A e. _V   =>   |- ((A e. Q. /\ B e. Q.) -> (A <Q B <-> E.x(x e. Q. /\ (A +Q x) = B)))
Theorem	halfpq 6234	One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120.
|- (A e. Q. -> E.x(x +Q x) = A)
Theorem	nsmallpq 6235	The is no smallest positive fraction.
|- (A e. Q. -> E.x x <Q A)
Theorem	ltbtwnpq 6236	There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120.
|- A e. _V   &   |- B e. _V   =>   |- (A <Q B -> E.x(A <Q x /\ x <Q B))
Theorem	ltrpq 6237	Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120.
|- A e. _V   &   |- B e. _V   =>   |- (A <Q B -> (*Q` B) <Q (*Q` A))
Definition	df-np 6238	Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of [Gleason] p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c 6392, and is intended to be used only by the construction.)
|- P. = {x | (((/) C. x /\ x C. Q.) /\ A.y e. x (A.z(z <Q y -> z e. x) /\ E.z e. x y <Q z))}
Definition	df-1p 6239	Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 6392, and is intended to be used only by the construction. Definition of [Gleason] p. 122.
|- 1P = {x | x <Q 1Q}
Definition	df-plp 6240	Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 6392, and is intended to be used only by the construction. From Proposition 9-3.5 of [Gleason] p. 123.
|- +P. = {<.<.x, y>., z>. | ((x e. P. /\ y e. P.) /\ z = {w | E.v e. x E.u e. y w = (v +Q u)})}
Definition	df-mp 6241	Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 6392, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124.
|- .P. = {<.<.x, y>., z>. | ((x e. P. /\ y e. P.) /\ z = {w | E.v e. x E.u e. y w = (v .Q u)})}
Definition	df-ltp 6242	Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 6392, and is intended to be used only by the construction. From Proposition 9-3.2 of [Gleason] p. 122.
|- <P = {<.x, y>. | ((x e. P. /\ y e. P.) /\ x C. y)}
Theorem	npex 6243	The class of positive reals is a set.
|- P. e. _V
Theorem	elnp 6244	Membership in positive reals.
|- (A e. P. <-> (((/) C. A /\ A C. Q.) /\ A.x e. A (A.y(y <Q x -> y e. A) /\ E.y e. A x <Q y)))
Theorem	prn0 6245	A positive real is not empty.
|- (A e. P. -> A =/= (/))
Theorem	prpssnq 6246	A positive real is a subset of the positive fractions.
|- (A e. P. -> A C. Q.)
Theorem	elprpq 6247	A positive real is a set of positive fractions.
|- ((A e. P. /\ B e. A) -> B e. Q.)
Theorem	0npr 6248	The empty set is not a positive real.
|- -. (/) e. P.
Theorem	prcdpq 6249	A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121.
|- ((A e. P. /\ B e. A) -> (C <Q B -> C e. A))
Theorem	prub 6250	A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122.
|- (((A e. P. /\ B e. A) /\ C e. Q.) -> (-. C e. A -> B <Q C))
Theorem	prnmax 6251	A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121.
|- ((A e. P. /\ B e. A) -> E.x(x e. A /\ B <Q x))
Theorem	prnmadd 6252	A positive real has no largest member. Addition version.
|- B e. _V   =>   |- ((A e. P. /\ B e. A) -> E.x(B +Q x) e. A)
Theorem	ltrelpr 6253	Positive real 'less than' is a relation on positive reals.
|- <P C_ (P. X. P.)
Theorem	genpv 6254	Value of general operation (addition or multiplication) on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   =>   |- ((A e. P. /\ B e. P.) -> (AFB) = {f | E.gE.h((g e. A /\ h e. B) /\ f = (gGh))})
Theorem	genpelv 6255	Membership in value of general operation (addition or multiplication) on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- C e. _V   =>   |- ((A e. P. /\ B e. P.) -> (C e. (AFB) <-> E.fE.g((f e. A /\ g e. B) /\ C = (fGg))))
Theorem	genpprecl 6256	Pre-closure law for general operation on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   =>   |- ((A e. P. /\ B e. P.) -> ((C e. A /\ D e. B) -> (CGD) e. (AFB)))
Theorem	genpdm 6257	Domain of general operation on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   =>   |- dom F = (P. X. P.)
Theorem	genpn0 6258	The result of an operation on positive reals is not empty.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   =>   |- ((A e. P. /\ B e. P.) -> (/) C. (AFB))
Theorem	genpss 6259	The result of an operation on positive reals is a subset of the positive fractions.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- ((g e. Q. /\ h e. Q.) -> (gGh) e. Q.)   =>   |- ((A e. P. /\ B e. P.) -> (AFB) C_ Q.)
Theorem	genpnnp 6260	The result of an operation on positive reals is different from the set of positive fractions.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- ((w e. Q. /\ v e. Q.) -> (wGv) e. Q.)   &   |- (z e. Q. -> (x <Q y <-> (zGx) <Q (zGy)))   &   |- (xGy) = (yGx)   =>   |- ((A e. P. /\ B e. P.) -> -. (AFB) = Q.)
Theorem	genpcd 6261	Downward closure of an operation on positive reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (gGh) -> x e. (AFB)))   =>   |- ((A e. P. /\ B e. P.) -> (f e. (AFB) -> (x <Q f -> x e. (AFB))))
Theorem	genpnmax 6262	An operation on positive reals has no largest member.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- (v e. Q. -> (z <Q w <-> (vGz) <Q (vGw)))   &   |- (zGw) = (wGz)   =>   |- ((A e. P. /\ B e. P.) -> (f e. (AFB) -> E.x(x e. (AFB) /\ f <Q x)))
Theorem	genpcl 6263	Closure of an operation on reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- ((x e. Q. /\ y e. Q.) -> (xGy) e. Q.)   &   |- (h e. Q. -> (f <Q g <-> (hGf) <Q (hGg)))   &   |- (xGy) = (yGx)   &   |- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (gGh) -> x e. (AFB)))   =>   |- ((A e. P. /\ B e. P.) -> (AFB) e. P.)
Theorem	genpass 6264	Associativity of an operation on reals.
|- F = {<.<.w, v>., u>. | ((w e. P. /\ v e. P.) /\ u = {x | E.y e. w E.z e. v x = (yGz)})}   &   |- B e. _V   &   |- C e. _V   &   |- dom F = (P. X. P.)   &   |- ((f e. P. /\ g e. P.) -> (fFg) e. P.)   &   |- ((fGg)Gh) = (fG(gGh))   =>   |- ((AFB)FC) = (AF(BFC))
Theorem	plpv 6265	Value of addition on positive reals.
|- ((A e. P. /\ B e. P.) -> (A +P. B) = {x | E.yE.z((y e. A /\ z e. B) /\ x = (y +Q z))})
Theorem	mpv 6266	Value of multiplication on positive reals.
|- ((A e. P. /\ B e. P.) -> (A .P. B) = {x | E.yE.z((y e. A /\ z e. B) /\ x = (y .Q z))})
Theorem	dmplp 6267	Domain of addition on positive reals.
|- dom +P. = (P. X. P.)
Theorem	dmmp 6268	Domain of multiplication on positive reals.
|- dom .P. = (P. X. P.)
Theorem	1pr 6269	The positive real number 'one'.
|- 1P e. P.
Theorem	addclprlem1 6270	Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123.
|- (((A e. P. /\ g e. A) /\ x e. Q.) -> (x <Q (g +Q h) -> ((x .Q (*Q` (g +Q h))) .Q g) e. A))
Theorem	addclprlem2 6271	Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123.
|- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g +Q h) -> x e. (A +P. B)))
Theorem	addclpr 6272	Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123.
|- ((A e. P. /\ B e. P.) -> (A +P. B) e. P.)
Theorem	mulclprlem 6273	Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124.
|- ((((A e. P. /\ g e. A) /\ (B e. P. /\ h e. B)) /\ x e. Q.) -> (x <Q (g .Q h) -> x e. (A .P. B)))
Theorem	mulclpr 6274	Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124.
|- ((A e. P. /\ B e. P.) -> (A .P. B) e. P.)
Theorem	addcompr 6275	Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123.
|- A e. _V   &   |- B e. _V   =>   |- (A +P. B) = (B +P. A)
Theorem	addasspr 6276	Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123.
|- B e. _V   &   |- C e. _V   =>   |- ((A +P. B) +P. C) = (A +P. (B +P. C))
Theorem	mulcompr 6277	Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124.
|- A e. _V   &   |- B e. _V   =>   |- (A .P. B) = (B .P. A)
Theorem	mulasspr 6278	Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124.
|- B e. _V   &   |- C e. _V   =>   |- ((A .P. B) .P. C) = (A .P. (B .P. C))
Theorem	distrlem1pr 6279	Lemma for distributive law for positive reals.
Theorem	distrlem2pr 6280	Lemma for distributive law for positive reals.
Theorem	distrlem3pr 6281	Lemma for distributive law for positive reals.
Theorem	distrlem4pr 6282	Lemma for distributive law for positive reals.
Theorem	distrlem5pr 6283	Lemma for distributive law for positive reals.
Theorem	distrpr 6284	Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124.
|- B e. _V   &   |- C e. _V   =>   |- (A .P. (B +P. C)) = ((A .P. B) +P. (A .P. C))
Theorem	1idpr 6285	1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124.
|- (A e. P. -> (A .P. 1P) = A)
Theorem	ltprord 6286	Positive real 'less than' in terms of proper subset.
|- ((A e. P. /\ B e. P.) -> (A <P B <-> A C. B))
Theorem	psslinpr 6287	Proper subset is a linear ordering on positive reals. Part of Proposition 9-3.3 of [Gleason] p. 122.
|- ((A e. P. /\ B e. P.) -> (A C. B \/ A = B \/ B C. A))
Theorem	ltsopr 6288	Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122.
|- <P Or P.
Theorem	prlem934a 6289	Sublemma for Lemma 9-3.4 of [Gleason] p. 122.
|- B e. _V   =>   |- (C e. N. -> (((B e. Q. /\ A.x(x e. A -> (x +Q B) e. A)) /\ y e. A) -> (y +Q ([<.C, 1o>.] ~Q .Q B)) e. A))
Theorem	prlem934b 6290	Sublemma for Lemma 9-3.4 of [Gleason] p. 122.
|- (((u e. N. /\ w e. N.) /\ (v e. N. /\ z e. N.)) -> (([<.(w .N v), 1o>.] ~Q .Q [<.z, w>.] ~Q ) = [<.v, u>.] ~Q \/ [<.v, u>.] ~Q <Q ([<.(w .N v), 1o>.] ~Q .Q [<.z, w>.] ~Q )))
Theorem	prlem934 6291	Lemma 9-3.4 of [Gleason] p. 122.
|- ((A e. P. /\ B e. Q.) -> E.x(x e. A /\ -. (x +Q B) e. A))
Theorem	ltaddpr 6292	The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123.
|- ((A e. P. /\ B e. P.) -> A <P (A +P. B))
Theorem	ltaddpr2 6293	The sum of two positive reals is greater than one of them.
|- B e. _V   =>   |- (C e. P. -> ((A +P. B) = C -> A <P C))
Theorem	ltexprlem1 6294	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem2 6295	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem3 6296	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem4 6297	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem5 6298	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem6 6299	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.
Theorem	ltexprlem7 6300	Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123.